Cissoid of Diocles

A cubic curve invented by Diocles in about 180 BC in connection with his attempt to duplicate the cube by geometrical methods. The name "cissoid" first appears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. From a given point there are either one or three tangents to the cissoid.

Given an origin and a point on the curve, let be the point where the extension of the line intersects the line and be the intersection of the circle of radius and center with the extension of . Then the cissoid of Diocles is the curve which satisfies .

The cissoid of Diocles is the roulette of a parabola vertex of a parabola rolling on an equal parabola. Newton gave a method of drawing the cissoid of Diocles using two line segments of equal length at right angles. If they are moved so that one line always passes through a fixed point and the end of the other line segment slides along a straight line, then the midpoint of the sliding line segment traces out a cissoid of Diocles.

The cissoid of Diocles is given by the parametric equations

			(1)
			(2)

for (Lawrence 1972, p. 99). Converting these to polar coordinates gives

(3)

As an implicit equation,

(4)

which is equivalent to

(5)

An alternate parametrization equivalent to that above is given by

			(6)
			(7)

(Yates 1952, p. 27).

The cissoid of Diocles has a cusp at the origin, and vertical asymptote at .

As found by Huygens and Wallis in 1658, the area between the curve and its vertical asymptote is

			(8)
			(9)

(MacTutor Archive).

In this parametrization, the arc length, curvature, and tangential angle are given by

			(10)
			(11)
			(12)

for and .

An alternative parametric form is

			(13)
			(14)

(Gray 1997) for . In this parametrization, the arc length, curvature, and tangential angle are

			(15)
			(16)
			(17)

for and .